To determine which equation models the same quadratic relationship as the given function [tex]\( f(x) = x^2 + 12x + 4 \)[/tex], we need to check if the given function can be rewritten in the form of the options presented. Let's start by rewriting [tex]\( f(x) \)[/tex] in a form that resembles the given choices.
1. Complete the square for [tex]\( f(x) = x^2 + 12x + 4 \)[/tex]:
The standard form of completing the square for a quadratic function [tex]\( ax^2 + bx + c \)[/tex] involves creating an expression that represents a perfect square trinomial. Here's the process:
[tex]\[
f(x) = x^2 + 12x + 4
\][/tex]
First, take the coefficient of [tex]\( x \)[/tex], which is 12, and divide it by 2:
[tex]\[
\frac{12}{2} = 6
\][/tex]
Next, square this result:
[tex]\[
6^2 = 36
\][/tex]
Add and subtract this square inside the function:
[tex]\[
f(x) = x^2 + 12x + 36 - 36 + 4
\][/tex]
Group the perfect square trinomial and simplify the constant term:
[tex]\[
f(x) = (x + 6)^2 - 36 + 4
\][/tex]
Simplify the constants:
[tex]\[
f(x) = (x + 6)^2 - 32
\][/tex]
2. Compare with the given options:
We can see that the equivalent form of [tex]\( x^2 + 12x + 4 \)[/tex] after completing the square is:
[tex]\[
f(x) = (x + 6)^2 - 32
\][/tex]
Now, we'll match this with the given options:
- Option A: [tex]\( y = (x + 6)^2 + 40 \)[/tex]
This is incorrect because it has +40 instead of -32.
- Option B: [tex]\( y = (x + 6)^2 - 32 \)[/tex]
This is correct because it matches our derived form exactly.
- Option C: [tex]\( y = (x \cdot 6)^2 - 32 \)[/tex]
This is incorrect because it involves [tex]\( x \cdot 6 \)[/tex] instead of [tex]\( x + 6 \)[/tex].
- Option D: [tex]\( y = (x - 6)^2 + 40 \)[/tex]
This is incorrect because it involves [tex]\( x - 6 \)[/tex] instead of [tex]\( x + 6 \)[/tex], and +40 instead of -32.
Therefore, the correct answer is:
[tex]\[ \boxed{B} \][/tex]
